How can complex numbers be expressed with a real denominator?

In summary, the student is trying to solve a homework problem in which they need to convert the denominator into the form a+ib. They are unsure how to do this and are grateful for the help of their tutor.
  • #1
patm95
31
0

Homework Statement


Express (see attachment) with a real denominator


Homework Equations



Not sure if there is really relevant equations to use here.

The Attempt at a Solution



First I multiply the top and bottom by the exponential. That gives me e^ix/(e^ix-r^2). I think this is a good first step, but I am unsure where to go from here.
 

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  • #2
To make the denominator real (called rationalizing the denominator) you need to convert it into the form a+ib and then multiply by its complex conjugate a-ib. Can you turn it into that form first? It will be easier if you convert eix into its polar form expression.
 
  • #3
If I convert the denominator as given in the problem I end up with: i r^2 sin(x)+r^2 (-cos(x))+1 I don't see how I can make a conjugate with 3 variables here? Am I missing something? I have went over this several times in attempt to make sure I hadn't made a mistake...
 
  • #4
I take it r and δ are real. If that's the case, it's a bit simpler to simply replace i with -i to form the conjugate of the denominator, rather than converting to rectangular form. Use the fact that [itex]z+\bar{z} = 2 \textrm{Re}(z)[/itex] to simplify the cross term.
 
  • #5
Whenever you have a problem of the form [tex]\frac{a}{b}[/tex] where 'a' and 'b' are in general, complex, you can immediately multiply top and bottom by the conjugate of b.

[tex]\frac{a}{b} \frac{b^*}{b^*}[/tex]

and the result in the denominator is b*b = |b|, which is always real.

If [tex]r, \delta[/tex] are real, the only change when you conjugate is changing the sign on the i's.
 
  • #6
You don't have to write the complex number as a+ bi. Just multiply both numerator and denominator by the complex conjugate, [itex]1- r^2e^{i\delta}[/itex]. You get
[tex]\frac{1- r^2e^{i\delta}}{1- r^2(e^{i\delta}+ e^{-i\delta})+ r^4}[/tex]
[tex]= \frac{1- r^2e^{i\delta}}{1+ r^4- 2r^2 cos(\delta)}[/tex]

(I see now that Vela had said essentially the same thing.)
 
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  • #7
Thanks very much for all the help. I did get this right thanks to all your help!
 

1. What is a complex number in the denominator?

A complex number in the denominator refers to a fraction in which the denominator is a complex number, meaning it contains both a real part and an imaginary part. For example, 1/(2+3i) is a complex number in the denominator.

2. Why are complex numbers in the denominator important?

Complex numbers in the denominator are important in mathematical calculations because they allow us to solve problems that involve both real and imaginary numbers. They also have applications in fields such as electrical engineering and physics.

3. How do you simplify a complex number in the denominator?

To simplify a complex number in the denominator, we use a process called rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the complex number in the denominator. The conjugate is the same number with the sign of the imaginary part changed. For example, to rationalize the denominator of 1/(2+3i), we would multiply both the numerator and denominator by (2-3i).

4. Can you divide by a complex number in the denominator?

Yes, you can divide by a complex number in the denominator. However, it is important to note that when dividing by a complex number, we must rationalize the denominator first before performing the division.

5. How are complex numbers in the denominator used in real life?

Complex numbers in the denominator have various real-life applications, such as in electrical circuits and systems, signal processing, and quantum mechanics. They are also used in the study of periodic functions and in solving differential equations.

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